Assume the scale of the periods is time. Let's simultaneously release the two signals ##f## and ##g## with a phase difference ##\delta## and with periods ##T_f## and ##T_g##. If the two signals have different periods, then the length of ##h=f+g##'s period is dependent on how many "iterations" it takes for both signals to again have a phase difference ##\delta##. So if ##f## has a period ##T##, and ##g## has a period of ##1.1T##, then ##h## will have a period of ##10T##. This means that the ratio ##T_f/T_g= 1/1.1 = 10/11 ## represents the lengths of time periods needed (10 for ##f## & 11 for ##g##) it takes for the signals to match up.

Now, I am tempted to say that this ratio must be rational in general because a whole number of ##f## and ##g## periods must transpire in time for the two signals to get back to where they started. But I'm a bit unsure still.

So if ##f## has a period ##T##, and ##g## has a period of ##1.1T##, then ##h## will have a period of ##10T##. This means that the ratio ##T_f/T_g= 1/1.1 = 10/11 ## represents the lengths of time periods needed (10 for ##f## & 11 for ##g##) it takes for the signals to match up.

It must be ##f## needs ##11## and ##g## needs ##10,## for that they both spend ##11T.##

Now, I am tempted to say that this ratio must be rational in general because a whole number of ##f## and ##g## periods must transpire in time for the two signals to get back to where they started. But I'm a bit unsure still.

As was stated, your intuition is correct. However, to convince a mathematician, you will need to prove this more rigorously. The "if" is easily shown as you have done, but more generally:
Assume f(t+Tf) = f(t) and g(t+Tg) = g(t) for all t. If Tf/Tg = n/m, then define T = n Tg = m Tf. We now have f(t+T) + g(t+T) = f(t+m Tf) + g(t + n Tg) = f(t) + g(t), showin that f+g is periodic (note that n and m need to be coprime for T to be the actual period).

The more involved part is showing the only if, i.e, to show that f+g is not periodic if the ratio of their periods is irrational. Can you think of a way?

Well, let's assume two signals ##f## and ##g## are unleashed with some phase difference ##\delta##. Then if ##n/m## is an irrational number there never will be a whole number of periods ##nT_g## and ##mT_f## that are equal to each other => the signal ##h=f+g## will not repeat itself.

I guess this isn't a proof though. I'd have to somehow prove that if ##n/m## is irrational then ##h## will have a period of infinity or something, but I don't even know how to derive the relationship between ##h##'s period and that of ##f## and ##g##.

By the way, it's kind of interesting how the non-repeating decimals trait of irrational ##n/m## is carried over to ##h## never repeating itself either.

I don't think you're missing anything except a more formal construction. One way might be to suppose that the ratio of periods is irrational, but that the signal is periodic (with finite period T). Then show that a contradiction results.